6(1) Example of Factor Analysis of Correlations

Research Question

:

People spend time on different activities The dataset Timbugt.Dat shows average time spent by twenty eight population groups on 10 major activities The population groups were selected according to the following criteria Sex, Marital Status, Employment Status and Country. The research question is: What is the structure of relationships between activities, between population groups, between activities and population groups?

Methodology

:

Factor Analysis of Correlations (Since the data matrix is not a contingency table, factor analysis of correlations was chosen as data analytic technique in preference to factor analysis of correspondences).

Dataset

:

TIMBUGT.DAT
SYNTAX*
$RUN FACTOR
$FILES
PRINT = TIMBUGT.LST
DICTIN = TIMBUGT.DIC
DATAIN = TIMBUGT.DAT
$SETUP
FACTOR ANALYSIS OF TIME BUDGET DATA
BADDATA=MD1 -
ANALYSIS=(NOCRSP,CORR) -
PVARS=(V2-V11) -
IDVAR=V1 -
NFACT=4 -
PRINT=(CDICT,STAT,MATR,VFPR,OFPR) 

Note: All options set at default values

EXTRACT FROM COMPUTER OUTPUT

After filtering 28 cases read from the input data file
Records kept= 28 / 28 in file; Weight of principal indiv. obs. in set : 28.00

 

Core matrix 5 P# 1 ( multiplied by 10000 ) ========

  WORK TRAN HOUS CHIL SHOP PERS MEAL SLEE TELE LEIS
WORK 10000                  
TRAN 9322 10000                
HOUS -9074 -8691 10000              
CHIL -8701 -8089 8613 10000            
SHOP -6571 -5029 5010 5426 10000          
PERS -1110 -791 -349 1239 5931 10000        
MEAL -5102 -6492 3844 4841 -175 -1836 10000      
SLEE -5433 -7245 4628 3203 112 -1856 7085 10000    
TELE -565 -437 -2060 1216 2156 3216 3798 764 10000  
LEIS -1914 -1053 -1125 -1086 2363 736 28 868 -957 10000

Trace= .10000E+02

 
Eigenvalues       VAL(1)=       4.6913
 --------------------------------------------------------------------------------------------------------------
 | NO |ITER|  Eigenvalue | Percent |  Cumul  |*|   Histogram of eigenvalues
 --------------------------------------------------------------------------------------------------------------
 |  1 |  0 |    4.691324 |  46.913 |  46.913 |*|***************|***************|***************|***************
 |  2 |  1 |    1.874915 |  18.749 |  65.662 |*|***************|*********
 |  3 |  2 |    1.374044 |  13.740 |  79.403 |*|***************|***
 |  4 |  2 |    1.127411 |  11.274 |  90.677 |*|**************
 -------------------------------------------------------------
 |  5 |  2 |     .492555 |   4.926 |  95.603 |*|******
 |  6 |  2 |     .258000 |   2.580 |  98.183 |*|***
 |  7 |  2 |     .110703 |   1.107 |  99.290 |*|*
 |  8 |  2 |     .043163 |    .432 |  99.721 |*|*
 |  9 |  5 |     .026898 |    .269 |  99.990 |*|
 -----------------------------------------------
 | 10 |  2 |     .000978 |    .010 | 100.000 |*|
CORRELATIONS     Factors   FACTOR ANALYSIS OF TIME BUDGET DATA 
 

Table of principal cases factors

 ---*----*--------------*--------------*--------------*--------------*---------------*------
    | IPR |  QLT WEIG INR|  1#F COS2 CPF|  2#F COS2 CPF|  3#F COS2 CPF|  4#F COS2 CPF|
 ---*----*--------------*--------------*--------------*--------------*----------------*-----
   1|    1|  848    0  30|-1710  345  22| -843   84  14|-1829  395  87|  448   24   6|
   2|    2|  793    0  25| -111    2   0|-2225  696  94| -465   30   6|  678   65  15|
   3|    3|  914    0  89| 4053  661 125|-2392  230 109| -571   13   8| -507   10   8|
   4|    4|  781    0  32|-1726  330  23| -425   20   3|-1890  395  93|  570   36  10|
   5|    5|  830    0  68| 3379  598  87|-1465  112  41|-1512  120  59|  -22    0   0|
   6|    6|  900    0  32|-1433  228  16|-2067  475  81|-1117  139  32| -732   59  17|
   7|    7|  786    0  45| -385   12   1|-2968  695 168| -994   78  26|  137    1   1|
   8|    8|  986    0  28|-1226  192  11| 2092  560  83|-1319  223  45| -301   12   3|
   9|    9|  689    0  16|  262   15   1| 1423  450  39| -445   44   5|  900  180  26|
  10|   10|  982    0  73| 4205  859 135| 1342   88  34| -779   30  16| -343    6   4|
  11|   11|  984    0  30|-1182  164  11| 2176  555  90|-1500  264  58| -109    1   0|
  12|   12|  983    0  48| 3030  688  70| 1818  248  63| -631   30  10|  485   18   7|
  13|   13|  990    0  35|-1439  214  16| 2324  559 103| -698   50  13|-1270  167  51|
  14|   14|  686    0  28| 1004  130   8| 1334  230  34| -505   33   7|-1508  294  72|
  15|   15|  917    0  18|-2121  895  34|  150    4   0|   25    0   0|  295   17   3|
  16|   16|  985    0  25| -954  128   7| -187    5   1| 1199  203  37| 2148  650 146|
  17|   17|  943    0  61| 3823  851 111|  -48    0   0|  923   50  22| -852   42  23|
  18|   18|  889    0  20|-2044  737  32|  244   11   1| -553   54   8|  707   88  16|
  19|   19|  964    0  21|  490   41   2|  141    3   0| 1057  188  29| 2083  732 137|
  20|   20|  890    0  55|-3001  582  69| -715   33  10| 1405  128  51|-1509  147  72|
  21|   21|  662    0  13|  -70    1   0| -723  148  10| 1129  362  33| -728  150  17|
  22|   22|  948    0  19|-2111  843  34|  295   16   2|  627   74  10| -277   15   2|
  23|   23|  982    0  28| -980  123   7|  254    8   1| 1403  251  51| 2169  600 149|
  24|   24|  943    0  59| 3437  719  90|  507   16   5| 1789  195  83| -476   14   7|
  25|   25|  949    0  19|-2205  904  37|  261   13   1|  389   28   4| -153    4   1|
  26|   26|  946    0  24| 1501  336  17|  524   41   5| 1570  368  64| 1162  201  43|
  27|   27|  981    0  44|-2133  369  35| -514   21   5| 1715  239  76|-2080  351 137|
  28|   28|  961    0  13| -354   34   1| -314   27   2| 1574  673  64| -915  227  27|
 ---*-----*--------------*--------------*--------------*--------------*--------------*-------
    |     |  28.   0 1000|          1000|          1000|          1000|          1000|
 ---*-----*--------------*--------------*--------------*--------------*--------------*-------
 

Table of principal variables factors

 ---*----*--------------*--------------*--------------*--------------*---------------
    | JPR |  QLT WEIG INR|  1#F COS2 CPF|  2#F COS2 CPF|  3#F COS2 CPF|  4#F COS2 CPF|
 ---*----*--------------*--------------*--------------*--------------*---------------
   1|    2|  977  187 100| -973  946 202|   93    9   5| -133   18  13|   70    5   4|
   2|    3|  966   36 100| -976  953 203|  -86    7   4|  -31    1   1|   67    4   4|
   3|    4|  988  115 100|  893  797 170|   53    3   1|  370  137 100|  227   51  46|
   4|    5|  881   14 100|  884  782 167| -106   11   6|   80    6   5|  286   82  73|
   5|    6|  908   45 100|  587  344  73| -744  554 295|   76    6   4|  -59    3   3|
   6|    7|  779   40 100|  114   13   3| -842  708 378| -240   57  42|   17    0   0|
   7|    8|  902   50 100|  644  415  88|  469  220 117| -509  259 188|  -90    8   7|
   8|    9|  797  327 100|  648  419  89|  510  261 139| -221   49  36| -261   68  61|
   9|   10|  913   41 100|  128   16   3| -252   63  34| -909  826 601|   84    7   6|
  10|   11|  956  144 100|   77    6   1| -196   38  20|  120   15  11| -947  897 796|
 ---*-----*--------------*--------------*--------------*--------------*--------------
    |     |     28.0 1000|          1000|          1000|          1000|          1000|
 ---*-----*--------------*--------------*--------------*--------------*--------------
 

Communalities

 Variable |Original final or.-fin |    1 |    2 |    3 |    4 |
 --------------------------------------------------------------
   1    2 |  .9771  .9771  .9E-06 | -920 |  114 |  291 |  179 |
   2    3 |  .9657  .9657  .8E-06 | -858 |   31 |  461 |  124 |
   3    4 |  .9880  .9880  .1E-05 |  973 |  137 | -113 |   92 |
   4    5 |  .8815  .8815  .5E-06 |  884 | -161 | -206 |  176 |
   5    6 |  .9076  .9076  .6E-06 |  647 | -620 |  198 | -253 |
   6    7 |  .7790  .7790  .8E-06 |  128 | -827 |  257 | -104 |
   7    8 |  .9021  .9021  .8E-06 |  291 |    0 | -902 |   53 |
   8    9 |  .7973  .7973  .7E-06 |  372 |  210 | -767 | -158 |
   9   10 |  .9133  .9133  .7E-06 | -195 | -754 | -516 |  199 |
  10   11 |  .9562  .9562  .1E-05 |  -20 |  -49 |  -44 | -975 |
INTERPRETATION

IDAMS reports that 28 cases were read from the input data file.

Weight of the principal individual observations is set equal to 28.

 

Core matrix of variables:
It is the matrix of correlation coefficients. All the coefficients are multiplied by 10,000.
Trace = Sum of the diagonal elements
           = Number of principal variables
           = 10 (since there are ten variables)

 

Table of Eigenvalues

Column 1

NO = Rank order of eigenvalues
Column 2 ITER = Number of iterations to obtain the computed value.
Column 3 Eigenvalue = l , each eigenvalue l corresponding to the factor a .
Column 4 Percent = Contribution of the factor to the total variance in percentage terms.
Column 5 Cumul = percentage of variance of factor axes, considered consecutively from the first factor axis.
Column 6

Histogram of eigenvalues = Each eigenvalue is represented by a line of asterisks (*).
The number of asterisks is proportional to the eigenvalue.
The first eigenvalue is always represented by 60 asterisks.

According to the criterion of average eigenvalue, only the first four factor axes each have eigenvalue ³ 1.00 Hence, the first four factors are taken for interpretation. These factors taken together explain more than 90% of the total variance.

 

Table of Principal Cases Factors

Column 1

NO = Serial number of the case
Column 2 IPR = Case Idcode
Column 3 QLT = quality of representation of the point i in the vector space, spanned by the first four factors.
Column 4 WEIG = Weight assigned to the point i. Each case is assigned equal weight. Since the total weight of all cases = 28, the weight of each case = 1.

The following columns are represented for each factor

# F       = Ordinate of the case on the factor axis
COS2  = Relative contribution of the factor axis to the eccentricity of the point i = Cosine squared of the angle formed by the vector radius i and the factor-axis.
CPF     = Relative contribution of the point i to the variance of the factor axis.
QLT indicates that all the row points are well represented in the vector space spanned by the first four axes.

  • Factor Axis – 1
  • (i) The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Rows: 3, 5, 10, 12, 17, 20, 24, 25

    (ii)

    The values of COS2 indicate that the following row points are well represented on this axis.

    Rows: 1, 3, 4, 5, 10, 12, 15, 17, 18, 20, 22, 24, 25, 26, 27

    Note that the following row points do not contribute significantly to the orientation of the factor axis, but they are reasonably well represented on this axis.

    Rows: 1, 4, 15, 18, 22, 26, 27

  • Factor Axis – 2
  • (i)

    The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Rows: 2, 3, 5, 6, 7, 8, 11, 12, 13

    (ii)

    The values of COS2 indicate that the following row points are well represented on this axis.

    Rows: 2, 6, 7, 8, 9, 11, 12, 13

    Note that rows 3 and 5 have more than average contribution to the orientation of the factor axis, but they are not represented on this axis. Similarly row 9 is well represented on this factor but it does not contribute to its orientation.

  • Factor Axis – 3
  • (i) The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Rows: 1, 4, 5, 8, 11, 23, 24, 26, 27, 28

    (ii)

    The values of COS2 indicate that the following row points are well represented on this axis.

    Rows: 1, 4, 11, 21, 23, 24, 26, 27, 28

    Note that rows 5 and 8 have more than average contribution to the orientation of the factor axis, but they are not represented on this axis. Similarly row 21 is well represented on this factor but it does not contribute much to its orientation.

  • Factor Axis – 4
  • (i)

    The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Rows: 13, 14, 16, 19, 20, 23, 27

    (ii) The values of COS2 indicate that the following row points are well represented on this axis.Rows: 14, 16, 19, 23, 27, 13

    Note that rows 13 have more than average contribution to the orientation of the factor axis, but it is not well represented on this axis.

 

Table of Principal Variables Factors

NO, QLT, INR, # F, CPF, COS2 have clearly been defined.

  • Factor Axis – 1
  • (i) The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Variables: V2, V3, V4, V5

    (ii)

    The values of COS2 indicate that the following row points are well represented on this axis.

    Variables: V2, V3, V4, V5

  • Factor Axis – 2
  • (i) The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Variables: V6, V7, V8, V9

    (ii) The values of COS2 indicate that the following row points are well represented on this axis.

    Variables: V6, V7, V9

  • Factor Axis – 3
  • (iii)

    The values of CPF indicate that the following rows have more than average contribution to the orientation of the factor axis.

    Variables: V8, V10

    (iv)

    The values of COS2 indicate that the following row points are well represented on this axis.

    Variables: V8, V10

  • Factor Axis – 4
  • (i)

    The values of CPF indicate that the following variable has more than average contribution to the orientation of the factor axis.

    Variable: V11

    (ii) The values of COS2 indicate that the following variable point is well represented on this axis.

    Variables: V11

 

Rotated Factors

The variable factors are rotated using the varimax procedure.

 

Table of communalties and Factor Loadings

Communalities
The communality shows how well a variable is predicted by the retained factors. It is similar to the R-Squared that would be obtained if this variable were regressed on the factors that were retained for interpretation.

Communality = Sum of squares of factor loadings
                       = Sum of COS squares
                       = Quality of representation of the variables in the factor space

It can be easily seen that the values of communality are exactly the same as those of QLT.

Factor loadings

Factor loadings are the correlations between the variables and factors. They are exactly equal to COSj .

  • Factor 1

The following variables have high loadings on this factor: V2, V3, V4, V5.

It is a bipolar factor: The poles are:

Positive: V4, V5, V6
Negative: V2, V3

  • Factor 2

The following variables have high loadings on this factor: V6, V7, V10.

It is a unipolar factor.

  • Factor 3

The following variables have high loadings on this factor: V8, V9, V10

It is a unipolar factor.

  • Factor 4

The following variables have high loadings on this factor: V6, V10

It is a unipolar factor.

 

Table 1
Contribution of explicative points to the composition of the first four factor axes
(Absolute contribution, multiplied by 1000)

Cloud Explicative points with positive coordinates Explicative points with negative coordinates
     
Axis 1 ( l1 = 4.691324, t 1 = 46.913)
Activities
V4, V5
V2, V3
Population Groups
3, 10, 12, 17, 24
20, 23
     
Axis 2 ( l2 = 1.874915, t 2 = 18.749)
Activities

V8, V9

V6, V7
Population Groups
8, 11, 12, 13
2, 3, 5, 6, 7
     
Axis 3 ( l3 = 1.374044, t 3 = 13.740)
Activities  
V8, V10
Population Groups
20, 23, 24, 26, 27, 28
1, 4, 5, 8, 11
     
Axis 4 ( l4 = 1.127411, t 4 = 11.274)
Activities  
V11
Population Groups
15, 19, 23
14, 20, 27, 28

 


Table 2
Contribution of the explained points to the eccentricities of the first four factorial axes
(Relative contribution, multiplied by 1000)


Cloud Explicative points with positive coordinates Explicative points with negative coordinates
     
Axis 1 ( l1 = 4.691324, t 1 = 46.913)
Activities V4, V5, V6 V2, V3
Population Groups 3, 5, 10, 12, 17, 24, 26 15, 18, 20, 22, 24, 27
     
Axis 2 ( l2 = 1.874915, t 2 = 18.749)
Activities V9 V6, V7
Population Groups 8, 9, 11, 13 2, 3, 6, 7
     
Axis 3 ( l3 = 1.374044, t 3 = 13.740)
Activities   V8, V10
Population Groups 21, 23, 26, 27, 28 1, 4, 11, 27
     
Axis 4 ( l4 = 1.127411, t 4 = 11.274)
Activities   V11
Population Groups 16, 19, 23 14

6(2) Example of Correspondence Analysis

Research Question

:

India has scientific cooperation links with several countries. The data (COOP.DAT) shows the number of coauthorship links with 35 most important partner countries in eleven scientific fields. The research question is: What is the structure of multivariate relationships between India’s significant partner countries and eleven scientific fields of cooperation?

Methodology

:

Factor Analysis of Correspondences

Dataset

:

COOP.DAT
SYNTAX
$RUN FACTOR
$FILES
PRINT = CORRES.LST
DICTIN = COOP.DIC
DATAIN = COOP.DAT
$SETUP
TRANSNATIONAL LINKS OF INDIAN SCIENCE
BADDATA=MD1 -
 ANALYSIS=(CRSP) -
 PVARS=(V2-V12) -
 IDVAR=1-
 NFACT=4 -
 PRINT=(CDICT,STAT,VFPR,OFPR)
 PLOT=USER
 X=1 Y=2 varp=(rincipal,suppl)
 X=3 Y=4 varp=(principal,supp)
 
EXTRACT FROM COMPUTER OUTPUT

After filtering 35 cases read from the input data file

Records kept= 35 / 35 in file ; Weight of principal indiv. obs. in set : 30.00

 
Variable RNK  Min  Max    Mean  Std.dev*  C.Var. Total  Variance* Skewness Kurtosis Weighted N
 ---------------------------------------------------------------------------------------------
 2  v2:M   1  .00  97.00   7.90   18.91    2.39   237.00   357.6104  3.7071  13.9348   30.00
 3  V3:P   2  .00 832.00 107.93  169.89    1.57  3238.00 28863.0300  2.7657   8.4335   30.00
 4  V4:C   3  .00 290.00  28.10   55.46    1.97   843.00  3075.2650  3.5684  13.7154   30.00
 5  V5:B   4  .00 120.00  13.40   25.53    1.91   402.00   651.8345  2.7428   7.7857   30.00
 6  V6:E   5  .00 132.00  13.80   26.04    1.89   414.00   678.3035  3.2107  11.3700   30.00
 7  V7:A   6  .00  58.00   6.63   12.69    1.91   199.00   160.9299  2.5946   6.8528   30.00
 8  V8:C   7 1.00 305.00  29.80   62.84    2.11   894.00  3948.8550  3.2791  10.5297   30.00
 9  V9:B   8 2.00 313.00  26.50   59.35    2.24   795.00  3522.1210  3.8172  15.2267   30.00
10  V10:   9  .00 188.00  16.13   35.62    2.21   484.00  1269.0160  3.8011  15.1668   30.00
11  V11:  10  .00  49.00   3.00    9.04    3.01    90.00    81.7241  4.4050  19.3863   30.00
12  V12:  11  .00  78.00   6.63   15.46    2.33   199.00   238.9988  3.4302  12.4631   30.00
 

Core matrix 1 P# 1 ( multiplied by 1000 )

  v2:M V3:P V4:C V5:B V6:E V7:A V8:C V9:B V10: V11:
v2:M 49                  
V3:P 109 467                
V4:C 55 208 128              
V5:B 40 134 81 68            
V6:E 41 147 79 57 64          
V7:A 28 88 51 51 38 70        
V8:C 58 207 115 83 80 59 139      
V9:B 58 196 109 79 76 58 116 117    
V10: 54 154 82 59 59 40 91 85 77  
V11: 26 64 38 27 25 15 37 38 32 20
Trace= .11984E+01                  
 
Trace=  .11984E+01
Eigenvalues       VAL(1)=       1.0000
 --------------------------------------------------------------------------------------------------------------
 | NO |ITER|  Eigenvalue | Percent |  Cumul  |*|   Histogram of eigenvalues
 --------------------------------------------------------------------------------------------------------------
    2   0       .079962    40.305      40.305 * *************** *************** *************** ***************
    3   1       .038363    19.337      59.641 * *************** **************
    4   2       .027867    14.047      73.688 * *************** ******
    5   2       .019775     9.968      83.656 * *************** 
 ----------------------------------------------
    6   5       .009519     4.798      88.453 * *******
    7   2       .009286     4.680      93.134 * *******
 ----------------------------------------------
    8   2       .006074     3.062      96.195 * *****
    9   3       .004243     2.139      98.334 * ***
   10   1       .003305     1.666     100.000 * **
-----------------------------------------------
 
Table of principal cases factors             
 ---*----*--------------*--------------*--------------*--------------*--------------*
     IPR |  QLT WEIG INR| 1#F COS2 CPF | 2#F COS2 CPF | 3#F COS2 CPF | 4#F COS2 CPF |
 ---*----*--------------*--------------*--------------*--------------*--------------*
   1     1  875  314  57  123  423  60  -123  420 123    29   24  10   -17    8   4 
   2     2  913  120  60  147  218  32    61   37  12  -132  175  75   219  484 292 
   3     3  543  109  14  -55  121   4    86  297  21   -54  118  12   -13    7   1 
   4     4  917   64  65   68   23   4  -157  123  41   382  724 334   -97   47  31 
   5     5  874   53  31 -260  580  45   120  123  20   -59   30   7  -128  141  44 
   6     6  879   55  44  172  184  20   -70   30   7  -248  384 121  -213  282 125 
   7     7  926   48  78 -526  863 167   132   54  22    51    8   5    10    0   0 
   8     8  771   26  59 -541  646  94   167   62  19   -75   12   5  -152   51  30 
   9     9  767   24  50  416  418  52   324  254  66   -65   10   4  -189   86  43 
  10    10  821   23  28 -356  522  37    63   17   2   111   51  10   238  233  66 
  11    11  770   21  23 -132   83   5     9    0   0   380  687 111    -5    0   0 
  12    12  917   16  28  160   74   5  -121   43   6  -246  175  34   464  625 173 
  13    13  912   14  32 -594  805  63   182   76  12  -111   28   6    32    2   1 
  14    14  937   14  28 -533  699  49   247  150  22    69   12   2   176   76  22 
  15    15  218   12  10  -59   21   1   -25    4   0   -77   37   2  -159  156  15 
  16    16  906   11  10 -392  829  21    46   12   1    29    5   0  -106   61   6 
  17    17  719   10  14 -406  586  20    68   16   1   133   63   6   122   53   7 
  18    18  714    8  28  356  183  12  -393  223  32   419  254  50   193   54  15 
  19    19  574    7  28  323  123   9  -330  128  19  -437  225  45  -288   98  28 
  20    20  698    6  17  220   87   4  -294  156  14  -497  447  55   -67    8   1 
  21    21  942    6  23 -776  800  47   318  135  16    52    4   1    55    4   1 
  22    22  187    6   5  135  108   1   -67   27   1   -76   34   1   -55   18   1 
  23    23  644    4  20 -105   13   1    61    4   0  -539  333  47  -507  294  58 
  24    24  478    5  20  386  180   9   443  236  24   195   46   6   116   16   3 
  25    25  997    5 190 1916  462 218   200    0 503   494  494  31    42  -96  12 
  26    26  870    4   7 -513  812  14   132   54   2   -14    1   0   -30    3   0 
  27    27  138    4   6  160   95   1   -61   14   0    66   16   1    61   14   1 
  28    28  451    3  18  359  120   5  -468  204  19   -65    4   0   362  122  22 
  29    29  433    3   3  100   51   0   -98   48   1   242  297   7    84   36   1 
  30    30  241    4   5  -92   33   0  -151   89   2    73   21   1   158   98   5 
 ---*----*--------------*--------------*--------------*--------------*--------------*
           7596.0 1000             1000           1000           1000           1000 
 ---*----*--------------*--------------*--------------*--------------*--------------*
 
Table of supplementary cases factors         
 ---*----*--------------*--------------*--------------*--------------*--------------
    |ISUP| QLT WEIG INR | 1#F COS2 CPF | 2#F COS2 CPF | 3#F COS2 CPF | 4#F COS2 CPF|
 ---*----*--------------*--------------*--------------*--------------*--------------
  1    31  608    4  15   619  466  17  -165   33   3   -99   12   1   281   96  14 
  2    32  648    3  21   706  378  20   165   21   2  -173   23   3   547  227  48 
  3    33  614    3  11  -573  516  13   196   60   3    75    9   1   136   29   3 
  4    34  432    3  24  -578  231  14   -59    2   0   480  160  27   239   40  10 
  5    35  973    3  22 -1036  797  44   455  154  18   140   15   2   101    8   2 
 ---*----*--------------*--------------*--------------*--------------*--------------
  |       7596.0     93 |          108 |           26 |           35 |           76|
 ---*----*--------------*--------------*--------------*--------------*--------------
 
Table of principal variables factors         
 ---*----*--------------*--------------*--------------*--------------*--------------*
 NO.  JPR  QLT WEIG INR   1#F COS2 CPF   2#F COS2 CPF   3#F COS2 CPF   4#F COS2 CPF 
 ---*----*--------------*--------------*--------------*--------------*--------------*
  1    2   899   31  91   116   23   5  -330  188  89   616  653 424  -144   36  33 
  2    3   998  426 204  -293  906 457    89   84  88    23    6   8    14    2   4 
  3    4   851  111  87    90   52  11  -102   67  30  -247  391 242  -230  341 298 
  4    5   689   53  74   397  570 104    84   26  10   -80   23  12  -140   71  52 
  5    6   145   55  47    83   40   5   -22    3   1   -61   22   7  -116   80  37 
  6    7   984   26 219   872  459 249   892  481 544   268   44  68    -6    0   0 
  7    8   949  118 106   224  281  74  -113   72  39  -144  115  87   293  481 512 
  8    9   433  105  64   219  393  63   -66   36  12   -19    3   1    10    1   1 
  9   10   772   64  67   170  139  23  -271  353 122   226  246 117    84   34   23 
  10  11   585   12  42   233   78   8  -460  303  65   274  108  32  -259   96   40 
 ---*----*--------------*--------------*--------------*--------------*--------------*
          7596.0    1000           1000           1000           1000            1000 
 ---*----*--------------*--------------*--------------*--------------*--------------*
 
Table of supplementary variables factors     
 ---*----*--------------*--------------*--------------*--------------*--------------*
     JSUP  QLT WEIG INR   1#F COS2 CPF   2#F COS2 CPF   3#F COS2 CPF   4#F COS2 CPF  
 ---*----*--------------*--------------*--------------*--------------*--------------*
  1   12    60   26  39    -5    0   0   -73   18   4   -91   28   8   -62   13   5  
 ---*----*--------------*--------------*--------------*--------------*--------------*
INTERPRETATION

IDAMS reports that the input data comprises:
35 rows
11 columns
5 rows and 1 column variables were treated as supplementary elements.

  Summary statistics of column variables
 

Core matrix. This is a matrix of relationships between principal variables. The elements (Cjj’) of the matrix are calculates as follows:

where Pi. = sum of principal variable
           Pj. = Sum of the principal case values

 

Table of Eigenvalues

Trace = Sum of non-trivial eigenvalues

= 1.1984 is the total inertia (variance) of the data matrix.

No. of Eigenvalues = No. of active column variables – 1 = 10-1=9

There are nine factorial axes.

The first axis has the highest eigenvalue and the last axis has the lowest eigenvalue.

NO = Serial number of eigenvalues in decreasing order

ITER = Number of iterations to obtain the factor axes, for each factor axis.

Eigenvalue = l a for a =2, …..10.

for a =1 l a =1.0

(trivial eigenvalue)

First Eigenvalue = 1.000 is trival and ignored in further analysis

Percent = Ta =Ta =100 l a /å (l i) = Percentage of variance of each factor axis.

Cumu = Cumulative percentage of variance of factor axes, considered consecutively from the first factor axis.

Histogram of eigenvalues = Each eigenvalue is represented by a line of asterisks (*). The number of asterisks is proportional to the eigenvalue.The first non-trivial eigenvalue is always represented by 60 asterisks.

According to the criterion of average eigenvalue, only the first four factor axes account for more than average eigenvalue (11 percent). Hence, these factors are taken for interpretation. All these factors together account for 83.656% of the total variance.

 

Table of cases factors

IPR = Serial number associatied with I (rows)

QLT = Quality of explanation of ith row in the factor space (1-4)

WEIG = Weight associated with ith row.

INR = Fraction of total inertia (variance) contributed by the ith row (multiplied by 1000).

The following columns are repeated for each factor – axis.

# F = The ordinate of a case on the factor axis (a =1, 2, 3, 4)

COS2 = Cosine squared of the angle formed by a case and the factor axis. It is a measure of the distance between a case and the factor. It is also equal to the relative contribution of the factor axis to the eccentricity of the point representing a case.

CPF = Relative contribution of a case to the variance of factor - axis.

The values of QLT indicate that rows 15, 22, 24, 27, 28, 29, 24 are poorly represented in the vector space spanned by the first four factor axes.

  • Factor Axis – 1
(i)

Values of CPF indicate that the following row points have more than average contribution to the orientation of the axis:
Rows: 1, 5, 7, 8, 9, 10, 13, 14, 21, 25

(ii)

Values of COS2 indicate that the following row points are well represented on this axis:
Rows: 1, 5, 7, 8, 9, 10, 13, 14, 16, 17, 21, 25, 26

Note that rows # 16, 17, 25, 26 are well represented on this axis, but they are not significant in terms of their contribution to the first axis.

  • Factor Axis – 2
(i)

Values of CPF indicate that the following rows have more than average contribution to the orientation of the axis:
Rows: 1, 4, 9, 25

(ii)

Values of COS2 indicate that the following row points are well represented on the axis:
Rows: 1, 3, 9, 25

  • Factor Axis – 3
(i)

Values of CPF indicate that the following rows are significant:
Rows: 2, 4, 6, 11, 18, 19, 20 23

(ii)

Values of COS2 indicate that the following row points are significant (COS2>.250):
Rows: 4, 6, 11, 18, 20

  • Factor Axis – 4
(i)

Values of CPF indicate that the following row points are significant:
Rows: 2, 5, 6, 9, 10, 12, 23

(ii)

Values of COS2 indicate that the following row points are significant (Cos2>.250):
Rows: 2, 6, 12, 23

 

Table of Supplementary Rows

QLT: Rows 34 is not well represented in the vector space spanned by first four axes.

INR: Supplementary rows do not contribute to the total inertia. The inertia of supplementary rows indicate whether the rows could play any role in the analysis if they were used as principal rows.

  • Factor Axis – 1
(i)

Values of COS2 indicate that the following row points are well represented on this axis.
Rows: 2, 6, 12, 23

  • Factor Axis – 2
(i)

Values of COS2 indicate that none of the supplementary row points are well represented on this axis.

  • Factor Axis – 3
(i)

Values of COS2 indicate that none of the supplementary row points are well represented on this axis.

(ii)  
  • Factor Axis – 4
(i)

Values of COS2 indicate that none of the supplementary row points are well represented on this axis.

(ii)  
 

Table of Principal Variables Factors

NO, QLT, INR, # F, CPF, COS2 have already been defined

QLT indicates that V6 is not well represent in the vector space spanned by the first for axes

  • Factor Axis – 1
(i)

Values of CPF indicate that the following column points (i.e. variables) are well represented on this axis.
Columns: 3, 5,7

(ii)

Values of COS2 indicate that the following points (i.e. variables) are well represented on this axis (COS2>.250):
Columns: 3, 5, 7, 8

  • Factor Axis – 2
(i)

Values of CPF indicate that the following row points are well represented on this axis.
Columns: 7, 10

(ii)

Values of COS2 indicate that the following row points are well represented on this axis (COS2>.250):
Columns: 7, 10, 11

  • Factor Axis – 3
(i)

Values of CPF indicate that the following row points are well represented on this axis.
Columns: 2, 4, 10

(ii)

Values of COS2 indicate that the following row points are well represented on this axis (COS2>.250):
Columns: 2, 4

  • Factor Axis – 4
(i)

Values of CPF indicate that the following row points are well represented on this axis.
Columns: 4, 8

(ii)

Values of COS2 indicate that the following row points are well represented on this axis (COS2>.250):
Columns: 4, 8

The above results are summarized in the following tables

   

Table 1

Contribution of explicative points to the composition of the first four factor axes

(Absolute contribution, multiplied by 1000)


Cloud Explicative points with positive coordinates Explicative points with negative coordinates

Axis 1 ( l1 = .079962, t 1 = 40.305)

Column Variables V5, V7 V3

Rows 1, 9, 25 5, 7, 8, 10, 13, 14, 16, 17, 21, 26

Axis 2 ( l2 = .038363, t 2 = 19.337)

Column Variables V7 V10

Rows 9, 25 1, 4

Axis 3 ( l3 = .027867, t 3 = 14.047)

Column Variables V2, V10 V4

Rows 4, 11, 18 2, 6, 9, 20, 23

Axis 4 ( l4 = .019775, t 4 = 9.968)

Column Variables V8 V4

Rows 2, 10, 12 5, 6, 9, 23


Table 2

Contribution of the explained points to the eccentricities of the first four factorial axes

(Relative contribution, multiplied by 1000)


Cloud Explained points with positive coordinates Explained points with negative coordinates

Axis 1 ( l1 = .079962, t 1 = 40.305)

Column Variables V5, V7, V8 V3

Rows 1, 9, 25, 31, 32 5, 7, 8, 10, 13, 14, 16, 17, 21, 26, 30, 35

Axis 2 ( l2 = .038363, t 2 = 19.337)

Column Variables V7 V10, V11

Rows 3, 9, 25 1

Axis 3 ( l3 = .027867, t 3 = 14.047)

Column Variables V2 V4

Rows 4, 11, 18, 23, 29 6, 20

Axis 4 ( l4 = .019775, t 4 = 9.968)

Column Variables V8 V4

Rows 2, 12 6, 23